Quadratic Formula and Discriminant
Real or complex roots, discriminant sign, and vertex of the parabola for ax^2 + bx + c = 0.
Example
You enter
- a (quadratic coefficient) 1
- b (linear coefficient) -3
- c (constant) 2
You get
- Root kind real-distinct
- Discriminant (b² - 4ac) 1
- Vertex (x, y) 1.5
- Vertex y -0.25
Details, formula, and sources
ax^2 + bx + c = 0 => x = (-b +/- sqrt(b^2 - 4ac)) / (2a). Discriminant D = b^2 - 4ac signs the root type: D > 0 two real, D = 0 one real double, D < 0 complex conjugate pair. Vertex at x = -b/(2a).
Classical algebra; mathematical fact. The closed form is attributed to Indian, Greek, and later Persian / European mathematicians; the modern algebraic form is universal.
No code citation required.
Estimate only. Readability formulas and similar metrics are derived from a representative population and have known edge-case noise. The classroom teacher governs final text selection, grade placement, and assessment decisions.
Field names used by the API: a, b, c, kind, discriminant, vertex_x, vertex_y
- Coefficients all three a, b, c are realconvention; complex coefficients are out of scope